Encyclopedia of Microtonal Music Theory

6mu / hexamu / 768-edo

[Joe Monzo, Tonalsoft Encyclopedia of Microtonal Music Theory]

A term coined in July 2003 by a group of tuning theorists (including Aaron Hunt, Gene Ward Smith, and Joe Monzo), to describe one of a family of terms referring to units of resolution in MIDI tuning, used in electronic music software and computer music software. The prefix specifies the exponent of 2 which describes the number of MIDI tuning units per semitone, and the final "mu" is an acronym for "MIDI unit". In this work the numerical figure is used in preference to the verbal prefix.

At the setting for 6mu pitch-bend resolution, a semitone is divided into 26 = 64 pitch-bend units. Thus there are 64 * 12 = 768 6mus in an "octave", so the 6mu measurement system may be thought of as 768-edo tuning, with a 6mu being one degree in 768-edo.

A 6mu is calculated as the 768th root of 2 -- 768√2, or 2(1/768) -- with a ratio of approximately 1:1.000902943. It is an irrational number, but is extremely close to the ratio 2217:2215 ( 31 5-1 443-1 7391 ):: the difference is ~ 1/70,000 of a cent, which for all intents and purposes makes the 6mu identical to that ratio. The formula for calculating the 6mu-value of any ratio is:
6mus = log10r * [ (26 * 12) / log10(2)]
or
6mus = log2r * (26 * 12)
, where r is the ratio.

A 6mu is:

exactly 25/64 (= 0.390625 = ~ 2/5) of a 300-edo savart

exactly 51/64 (= 0.796875 = ~ 4/5) of a 612-edo schisma

exactly 1 29/96 (= 1.30208333... = ~ 1 1/3) millioctaves

exactly 1 1/3 (= 1.3333... ) yamaha-units

exactly 1 9/16 (= 1.5625 = ~ 1 1/2) cents

exactly 3 13/64 (= 3.203125 = ~ 3 1/5) minas

exactly 11 91/768 (=
11.11848958,3... = ~ 11 1/9) tinas

exactly 13 77/96 (= 13.80208333... = ~ 13 4/5) türk-sents

exactly 39 151/768 (= 39.19661458333... = ~ 39 1/5) jots

approximately 48 (= ~ 47.95394297 = ~ 47 62/65) tuning-units

exactly 64 12mus

exactly 256 14mus

The internal data structure of the 6mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or -) showing the direction of the pitch-bend up or down, and all six of the remaining bits used for the tuning data, as follows:

let "d" designate the bits that cannot be used
because it is reserved for the SysEx flag, to
indicate that this is a byte of pitch-bend data.
let "s" designate the bit that represents the
sign of the pitch-bend data, + or - .
let "x" designate unused bits
the 6mu spec thus uses a total of 2+6 = 8 bits.
thus, the maximum possible value is:
ds11 1111 [binary]
= +/- 3F [hex]
= +/- 63 [decimal]
note that the first nibble can only indicate the sign + or -
and the data-values 0, 16, 32, or 48 [decimal].

For practical use in tuning MIDI-files, an interval's semitone value must first be calculated. The nearest integer semitone is translated into a MIDI note-number (which can generally also be described by letter-name plus optional accidental: A, Bb, C#, etc., followed by an "octave" register-number, as A-1, Bb2, etc.). Then the remainder or deficit is converted into 6mus plus or minus, respectively.

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6mus calculator

Ratio may be entered as fraction or floating-point decimal number.
(value must be greater than 1)

For EDOs (equal-temperaments), type: "a/b" (without quotes)
where "a" = EDO degree and "b" = EDO cardinality.
(value must be less than 1)

Enter ratio: = 6mus

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